# Covering complete graphs by monochromatically bounded sets

## TL;DR

This paper investigates a strengthened version of a classic graph covering problem, proving that for any 4-colouring of a complete graph, three monochromatic sets with bounded diameter can cover all vertices.

## Contribution

It establishes that in any 4-colouring of a complete graph, three monochromatic sets with bounded diameter suffice to cover all vertices, advancing understanding of monochromatic coverings.

## Key findings

- For 4-colourings, three monochromatic sets with diameter ≤ 160 cover all vertices.
- Strengthens the Lovász-Ryser conjecture by adding diameter constraints.
- Provides bounds on the diameter of monochromatic components needed for coverage.

## Abstract

Given a $k$-colouring of the edges of the complete graph $K_n$, are there $k-1$ monochromatic components that cover its vertices? This important special case of the well-known Lov\'asz-Ryser conjecture is still open. In this paper we consider a strengthening of this question, where we insist that the covering sets are not merely connected but have bounded diameter. In particular, we prove that for any colouring of $E(K_n)$ with 4 colours, there is a choice of sets $A_1, A_2, A_3$ that cover all vertices, and colours $c_1, c_2, c_3$, such that for each $i = 1,2,3$ the monochromatic subgraph induced by the set $A_i$ and the colour $c_i$ has diameter at most 160.